User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions
No edit summary |
No edit summary |
||
| Line 7: | Line 7: | ||
=== Basic Diatonic Functions === | === Basic Diatonic Functions === | ||
The three basic functions have their roots in [[LCJI]] | The three basic functions have their roots in [[LCJI]]. The functions themselves are labeled as follows: | ||
'''Tonic''' - This is the note that serves as the tonal center, and thus, the main resolution tone, and is the note for which scales are named (e.g. the key of C major is so-named because in this scale, C serves as the Tonic). This functionality has its roots in the fundamental at the root of both the harmonic and subharmonic series, which for all intents and purposes, can be thought of as [[1/1]], and, in [[octave equivalence|octave equivalent]] systems, [[2/1]]. Beyond being simply one of the primary three functions in German Theory, it is the only function that is known to be universal when it comes to tonal music, with the various other functions being collectively defined as ''Nontonic'', thus, it shouldn't come as a great surprise that the Tonic exerts a very powerful influence on the context of functional harmony regardless of the nature of the tonal music system in question, even helping to define aspects of the other primary functions on the chord level as opposed to the root level. In the realm of microtonality, the Tonic is not an interval that admits a lot of competition outside of modulation, and thus, deviations from a perfect 1/1 of up to 3.5 [[cent]]s away from the Tonic are considered here to be found in the Tonic's "event horizon", in which they are either absorbed into the bandwidth of the Tonic itself, altered through [[fudging]], or simply [[Tempering out|tempered out]] | '''Tonic''' - This is the note that serves as the tonal center, and thus, the main resolution tone, and is the note for which scales are named (e.g. the key of C major is so-named because in this scale, C serves as the Tonic). This functionality has its roots in the fundamental at the root of both the harmonic and subharmonic series, which for all intents and purposes, can be thought of as [[1/1]], and, in [[octave equivalence|octave equivalent]] systems, [[2/1]]. Beyond being simply one of the primary three functions in German Theory, it is the only function that is known to be universal when it comes to tonal music, with the various other functions being collectively defined as ''Nontonic'', thus, it shouldn't come as a great surprise that the Tonic exerts a very powerful influence on the context of functional harmony regardless of the nature of the tonal music system in question, even helping to define aspects of the other primary functions on the chord level as opposed to the root level. In the realm of microtonality, the Tonic is not an interval that admits a lot of competition outside of modulation, and thus, deviations from a perfect 1/1 of up to 3.5 [[cent]]s away from the Tonic are considered here to be found in the Tonic's "event horizon", in which they are either absorbed into the bandwidth of the Tonic itself, altered through [[fudging]], or simply [[Tempering out|tempered out]]. To use a character metaphor for how the Tonic acts in functional harmony, the Tonic is the king of the Kingdom of Tonality- a very good king who not only exercises the highest authority in matters of governing the kingdom and does not tolerate challenges to his leadership, but also knows how be a top-notch confidante to his subjects both wherever and whenever possible. | ||
'''Dominant''' - As per the name, and as noted on the [[Wikipedia:Dominant (music)|Wikipedia article]], the Dominant is the second most important after the Tonic, and, to use a character metaphor for how the Dominant acts in functional harmony, the Dominant is both the Head Steward of the Tonic's castle, and the one that executes the Tonic's directives as a Manager of Civil Service in the Kingdom of Tonality. However, in contrast to what is stated about the Dominant in the article, there are several caveats which must be addressed in the realm of microtonality. Firstly, one must take stock of the fact that, aside from the Unison and Octave, each octave-reduced harmonic and corresponding subharmonic interval come together to generate their own axis which has a preferred direction of travel<ref>[https://www.youtube.com/watch?v=HBdWxSxxe1M Quartertone Harmony - Beyond the Circle of Fifths SD 480p]</ref> which is determined by a Tonality's direction of construction. Secondly, one must take stock of the fact that when you take the notes that occur before the Tonic on each of these axes when moving in the preferred direction of travel and place them in a sequence, one finds that a clear hierarchy of functional strength based on the closeness of harmonic and subharmonic connection to the Tonic becomes apparent, with the [[3/2]] Perfect 5th away from the Tonic in the direction of tonality construction naturally emerging as the note with the strongest connection to the Tonic, though it should be noted that the relationships in this hierarchy are quite sensitive to detuning. Thus, the term "Dominant"- in its most basic form as referred to in this article, and specifically at the root level- is restricted to where it only refers to such notes that occur roughly at a 3/2 interval away from the Tonic in the scale's direction of construction, with acceptable detuning levels being at around 3.5 cents from JI on either side. With all that said, it should be noted that the level of importance typically associated with the Dominant goes instead to a different note instead of a 3/2 Perfect 5th in scales where the 5th scale degree is too far away from 3/2, and that there are a variety of other interval constructions which have the tendency to create tension which requires the Tonic to resolve- these surrogate Dominants are called "'''Paradominants'''". On the chord level, not only is the root level definition of the Dominant function at play, but it should also be noted that no Dominant or Paradominant sees the Tonic occurring in the proximal structure of its chord- that is, as a third or fifth. | '''Dominant''' - As per the name, and as noted on the [[Wikipedia:Dominant (music)|Wikipedia article]], the Dominant is the second most important after the Tonic, and, to use a character metaphor for how the Dominant acts in functional harmony, the Dominant is both the Head Steward of the Tonic's castle, and the one that executes the Tonic's directives as a Manager of Civil Service in the Kingdom of Tonality. However, in contrast to what is stated about the Dominant in the article, there are several caveats which must be addressed in the realm of microtonality. Firstly, one must take stock of the fact that, aside from the Unison and Octave, each octave-reduced harmonic and corresponding subharmonic interval come together to generate their own axis which has a preferred direction of travel<ref>[https://www.youtube.com/watch?v=HBdWxSxxe1M Quartertone Harmony - Beyond the Circle of Fifths SD 480p]</ref> which is determined by a Tonality's direction of construction. Secondly, one must take stock of the fact that when you take the notes that occur before the Tonic on each of these axes when moving in the preferred direction of travel and place them in a sequence, one finds that a clear hierarchy of functional strength based on the closeness of harmonic and subharmonic connection to the Tonic becomes apparent, with the [[3/2]] Perfect 5th away from the Tonic in the direction of tonality construction naturally emerging as the note with the strongest connection to the Tonic, though it should be noted that the relationships in this hierarchy are quite sensitive to detuning. Thus, the term "Dominant"- in its most basic form as referred to in this article, and specifically at the root level- is restricted to where it only refers to such notes that occur roughly at a 3/2 interval away from the Tonic in the scale's direction of construction, with acceptable detuning levels being at around 3.5 cents from JI on either side. With all that said, it should be noted that the level of importance typically associated with the Dominant goes instead to a different note instead of a 3/2 Perfect 5th in scales where the 5th scale degree is too far away from 3/2, and that there are a variety of other interval constructions which have the tendency to create tension which requires the Tonic to resolve- these surrogate Dominants are called "'''Paradominants'''". On the chord level, not only is the root level definition of the Dominant function at play, but it should also be noted that no Dominant or Paradominant sees the Tonic occurring in the proximal structure of its chord- that is, as a third or fifth. | ||
| Line 31: | Line 31: | ||
'''Imitation''' - These are notes that can substitute for either the Dominant or Serviant functions through chromatic-type alteration. Functions which have this kind of role are denoted with a "mock-" prefix here. | '''Imitation''' - These are notes that can substitute for either the Dominant or Serviant functions through chromatic-type alteration. Functions which have this kind of role are denoted with a "mock-" prefix here. | ||
'''Detempering''' - These are notes that appear when the comma or subchroma that separates them from the Tonic or from one of the primary Nontonic functions are not tempered out, and often, though not always, fall within the uncanny | '''Detempering''' - These are notes that appear when the comma or subchroma that separates them from the Tonic or from one of the primary Nontonic functions are not tempered out, and often, though not always, fall within the uncanny valleys of the three primary functions. | ||
== Facets Derived from Viennese Theory == | == Facets Derived from Viennese Theory == | ||
| Line 79: | Line 79: | ||
'''Subdislocant''' - This is a note that is to either the Lead or the Contralead what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone downwards and thus being displaced by a Nontonic function which leads away from the Tonic proper. | '''Subdislocant''' - This is a note that is to either the Lead or the Contralead what a Tyrant is to a Sycophant. Specifically, it is the result of the Tonic being altered by some kind of chromatic semitone downwards and thus being displaced by a Nontonic function which leads away from the Tonic proper. | ||
== | == Wolf Fifths, Wolf Fourths and the Uncanny Valleys of Harmony == | ||
As per Flora's later analysis<ref>''[[User:FloraC/Analysis on the 13-limit just intonation space: episode ii #Chapter V. Things Repel the Similar but Not Identical| Analysis on the 13-limit just intonation space: episode ii]]''. Flora Canou. Xenharmonic Wiki.</ref>, there are uncanny valleys on either side of each of the three primary functions. However, it seems that under certain circumstances, the average listener acquainted with [[12edo]] music will not notice wolf fourths or wolf fifths unless they are explicitly pointed out, thus there are not only additional functions hiding in the shadows of the three primary functions, but rules governing their usage, which shall be covered in this section. | |||
=== Circumtonic Regions === | === Circumtonic Regions === | ||
The Circumtonic regions are the two main regions on either side of the Tonic, outside the Tonic's "event horizon". These intervals are invariably inside the Tonic's uncanny valleys, and as | The Circumtonic regions are the two main regions on either side of the Tonic, outside the Tonic's "event horizon". These intervals are invariably inside the Tonic's uncanny valleys, and the uncanny valleys around the Tonic, unlike those around the other two functions, are relatively large, strong, and, most importantly, unconditional, and as a result, wolf intervals around the Tonic are only tolerable in melody and ornamentation but not harmony. Thus, these intervals cannot be directly approached, even melodically, and so they're usually avoided outside of modulation. | ||
'''Supercommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly | '''Supercommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 25 cents above the Tonic. These intervals are little more than stepping stones in modulation, and extra intervals that can be used together with the Tonic for a sense of dissonance, or for a slightly less resolved version of a Unison or Octave. | ||
'''Subcommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly | '''Subcommatic''' - This is a note that occurs at intervals from about 3.5 cents to roughly 25 cents below the Tonic. As with Supercommatic intervals, these intervals are little more than stepping stones in modulation, and extra intervals that can be used together with the Tonic for a sense of dissonance. | ||
=== Circumdominant Regions === | === Circumdominant Regions === | ||
| Line 95: | Line 95: | ||
The circumdomimant regions are the two main regions on either side of the Dominant proper, and since the uncanny valleys around the Dominant are not present unconditionally, there is more room for actual first derivative diatonic functionality, as well as other functionalities. | The circumdomimant regions are the two main regions on either side of the Dominant proper, and since the uncanny valleys around the Dominant are not present unconditionally, there is more room for actual first derivative diatonic functionality, as well as other functionalities. | ||
'''Geminodominant''' - This is a note that occurs roughly at intervals ranging from about | '''Geminodominant''' - This is a note that occurs roughly at intervals ranging from about 25 cents to about 3.5 cents short of the 3/2 perfect fifth in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Dominant through detempering, occurring in [[5-limit]] diatonic environments, and acting as a sort of "fraternal twin" to the Dominant, hence its name. Specifically, as typified by intervals like [[40/27]], Geminodominants are dissonant intervals that simultaneously act as alternatives to the Dominant in both chord progressions and chord construction, and often require resolution, though they also have a Preserviant function. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on the traditional weak harmonies- the Mediant and the Contramediant- which also has the benefits of strengthening interrupted cadences and creating the sense of impending movement, but outside of these usages and well supported chords, this kind of thing is best avoided since things repel the similar but not identical. Apart from diatonic contexts, Geminodominants only rise to prominence in systems where what might otherwise function as a Dominant is found just short of the sweet spot range near the standard issue 3/2. | ||
'''Pseudodominant''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents to about 25 cents beyond the 3/2 perfect fifth in the scale's direction of construction. Because none of these intervals arise naturally in a [[5-limit]] diatonic scale, and because they fall within one of the Dominant's uncanny valleys, it is rare to see this functionality outside of systems where what might otherwise function as a Dominant is found just beyond the sweet spot range near the standard issue 3/2. | |||
=== Circumserviant Regions === | === Circumserviant Regions === | ||
| Line 101: | Line 103: | ||
The circumserviant regions are the two main regions on either side of the Serviant proper, and since the uncanny valleys around the Serviant are not present unconditionally, there is more room for actual first derivative diatonic functionality, as well as other functionalities. | The circumserviant regions are the two main regions on either side of the Serviant proper, and since the uncanny valleys around the Serviant are not present unconditionally, there is more room for actual first derivative diatonic functionality, as well as other functionalities. | ||
'''Geminoserviant''' - This is a note that occurs roughly at intervals ranging from about | '''Geminoserviant''' - This is a note that occurs roughly at intervals ranging from about 3.5 cents to about 25 cents beyond the 4/3 perfect fourth in the scale's direction of construction. Although often overlooked or even outright shunned by traditional theorists, the Geminodominant is a legitimate diatonic function in terms of this analysis- albeit one only existing in non-meantone environments in which it is easily derived from the Serviant through detempering, occurring in 5-limit diatonic environments, and acting as a sort of "fraternal twin" to the Serviant, hence its name. As typified by intervals like [[27/20]], Geminoserviants are dissonant intervals that often act as a sort of predominant and or as the inverses of Geminodominants. The dissonance of this function relative to a chord root is useful in preventing tonicization of chords built on Mediant and the Contramediant, but outside of these usages and well supported chords, this kind of thing is best avoided. Apart from diatonic contexts, Geminoserviants only rise to prominence in systems where what might otherwise function as a Serviant is found just beyond the sweet spot range near the standard issue 4/3. | ||
'''Pseudoserviant''' - This is a note that occurs roughly at intervals ranging from about 25 cents to about 3.5 cents short of the 4/3 perfect fourth in the scale's direction of construction. Because none of these intervals arise naturally in a [[5-limit]] diatonic scale, and because they fall within one of the Dominant's uncanny valleys, it is rare to see this functionality outside of systems where what might otherwise function as a Dominant is found just short the sweet spot range near the standard issue 3/2. | |||
=== Governing Rules === | |||
'''Commatic Repulsion''' - This rule is best summed up in Flora Canou's statement that things repel the similar but not identical. To illustrate this principle, let's take a look at the VImin chord as is present in [[meantone]] and contrast it to the situation outside of meantone. Believe it or not, the meantone VImin chord has no fewer than three separate functions relative to the Tonic. First of all, there's the Relative Minor functionality, which is 5/3–1/1–5/4 relative to the Tonic and can tonicize as a new Imin either temporarily, as in a deceptive cadence, or permanently as in a modulation. Secondly, there's the Tertiary Dominant functionality, which is 27/16-81/80-81/64 relative to the Tonic and, as the name suggests, acts as the Dominant of the Dominant's own Dominant. Finally, there's the Tensive functionality, which is 27/16-1/1-5/4 relative to the Tonic and is involved in interrupted cadences, as well as in starting cadences that "wind down" such as VImin-IVmaj-Imaj or even VImin-Vmaj-Imaj. While all three of these are fused together in meantone, the first two of these three functions are separated exactly by the syntonic comma in non-meantone environments, while the third shares its root with the Tertiary Dominant and shares both its third and fifth with the Relative Minor, and the only way to use a VImin chord with all three functions is to temper out the syntonic comma. As Flora Canou has stated: | |||
<div style="font-style: italic; border: 1px solid silver; margin: 15px; padding: 15px;"> | |||
: If the comma is not tempered out, the progression does not hold. The idea to force it is absurd. Fitting one function will leave the other emergent function[s] misplaced by a comma, which is easily the most catastrophic scene [musically] – the uncanny valley of harmony. | |||
</div> | |||
However, it must be pointed out that there are caveats to this rule in regards to wolf fifths and wolf fourths, something which has been hinted at already and will be addressed in further detail below. | |||
'''Primary Adpositive Purity''' - This rule is that for every chord root located one step away from either the Tonic, Dominant or Serviant along the Circle of Fifths, there is a demand for the fifth of the chord in question to be within 3.5 cents of a just 3/2. This means that the Tonic, Dominant, Serviant, Supertonic and Subtonic chords all demand a perfect fifth as the fifth of the chord, whether you are building the Tonality upwards or downwards. One of the obvious applications of this is that chords built with wolf fifths must have roots located three or more steps away from the Tonic along the Circle of Fifths, and that when two notes within a given Diatonic system are separated by a wolf fifth, they both must likewise be located three or more steps away from the Tonic along the Circle of Fifths. | |||
== Paradiatonic and Parachromatic Functions == | == Paradiatonic and Parachromatic Functions == | ||
| Line 158: | Line 176: | ||
'''Neutral Contramediant''' - As per the name, this is nothing other than a neutralized Contramediant, and thus, it has the Tonic Parallel, Dominant Adjacent, Preserviant and Predominant functions that you expect from a Contramediant, but, like with the Neutral Mediant, it doesn't serve well at phrase endings, rather, its Tonic Parallel function is only appropriate during the middle of musical phrases. | '''Neutral Contramediant''' - As per the name, this is nothing other than a neutralized Contramediant, and thus, it has the Tonic Parallel, Dominant Adjacent, Preserviant and Predominant functions that you expect from a Contramediant, but, like with the Neutral Mediant, it doesn't serve well at phrase endings, rather, its Tonic Parallel function is only appropriate during the middle of musical phrases. | ||
'''Subgradient''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly | '''Subgradient''' - This is a note that occurs at intervals between roughly 48/25 above the Tonic and roughly 25 cents below the octave reduplication of the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, in Bass-Up Tonality, the Lead- however, there are functional differences between a Subgradient and a Lead in Bass-Up Tonality. In Bass-Up Tonality, Subgradients are often more likely to be passing tones than Leads, and, when they’re not merely passing non-chord tones, they are often harder to approach and or follow up without creating some kind of awkward tonal disconnect, with such a disconnect being especially noticeable for intervals like [[64/33]]. Furthermore, whereas a Lead can resolve to the Tonic in part through a strong harmonic connection, a Subgradient is much more likely to do so through sheer brute force, and even these cases require a proper set-up, as otherwise, the awkward tonal disconnect between the Subdietic and the Tonic is likely to result in the Subgradient resolving back down to either the Lead or the Semilead. As if that weren't enough, the Subgradient also has the Antintersubiant function. This function used to be called the "Subdietic", though that term has since been restricted to a related composite function (see below on Composite Functions). | ||
'''Supergradient''' - This is a note that occurs at intervals between roughly | '''Supergradient''' - This is a note that occurs at intervals between roughly 25 cents above the Tonic and 25/24 above the Tonic. These intervals tend to act as parachromatic alterations of either the Tonic, or, in Bass-Up Tonality, the Contralead. However, there there are a few functional differences between a Supergradient and a Contralead in Bass-Up Tonality that are worth considering. For starters, Supergradients are often more likely to be passing tones than Contraleads, and, when they’re not merely passing non-chord tones, they are just as liable to resolve upward thought some sort of semitone-like motion to some form of Contralead or Semicontralead, as they are to resolve downwards toward the Tonic, a property which intervals like 33/32 in particular are apt to demonstrate. Furthermore, whereas a Contralead can resolve to the Tonic in part through a strong harmonic connection, a Supergradient is much more likely to do so through sheer brute force when such a resolution is noticeable. As if that weren't enough, the Subgradient also has the Antinterregnant function. This function used to be called the "Superdietic", though that term has since been restricted to a related composite function (see below on Composite Functions). | ||
=== Basic Parachromatic Functions === | === Basic Parachromatic Functions === | ||
| Line 195: | Line 213: | ||
'''Contrabilancant''' - This function is a compositing of Contralead and Semicontralead functions, however, the tendency to resolve to the Tonic or to the Supertonic are about equal due to Tonic having greater gravity than the Supertonic. | '''Contrabilancant''' - This function is a compositing of Contralead and Semicontralead functions, however, the tendency to resolve to the Tonic or to the Supertonic are about equal due to Tonic having greater gravity than the Supertonic. | ||
These are composite functions immediately surrounding the Antitonic region | These are composite functions immediately surrounding the Antitonic region | ||